Integral points on the congruent number curve

We study integral points on the quadratic twists $\mathcal{E}_D:y^2=x^3-D^2x$ of the congruent number curve. We give upper bounds on the number of integral points in each coset of $2\mathcal{E}_D(\mathbb{Q})$ in $\mathcal{E}_D(\mathbb{Q})$ and show that their total is $\ll (3.8)^{\mathrm{rank} \mathcal{E}_D(\mathbb{Q})}$. We further show that the average number of non-torsion integral points in this family is bounded above by $2$. As an application we also deduce from our upper bounds that the system of simultaneous Pell equations $aX^2-bY^2=d$, $bY^2-cZ^2=d$ for pairwise coprime positive integers $a,b,c,d$, has at most $\ll (3.6)^{\omega(abcd)}$ integer solutions.